Separate phase space into areas of increasing and decreasing prey or predator abundance.
March 21, 2022
Separate phase space into areas of increasing and decreasing prey or predator abundance.
If isoclines are perpendicular … eternal oscillations.
If isoclines tilts … damped oscillations
On the left … Allee effect and Prey switching
On the right … Density dependence
You can send the system into an unstable state and INCREASE the probability of extinction.
“…increasing the supply of limiting nutrients or energy tends to destroy the steady state. Thus man must be very careful in attempting to enrich an ecosystem in order to increase its food yield. There is a real chance that such activity may result in decimation of the food species that are wanted in greater abundance.” - Rosenzweig (1971)
But is it real?
More experiments with Didinium nausutum vs. Paramecium caudatum.
Luckinbill 1973 recreates Gause experiments with a different food source.
When the food source is limited … some nice oscillations!
Harrison (1995) reinterprets the data
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efforts to show this effect generally fail. (reviewed in Roy and Chattopadhyay 2007).
Because things are complicated! Fluctuations are buffered by:
As long as there is some predator free areas, the oscillations are stabilized.
Example: Wildlife management “stumbled” in to this with wildlife reserves, the National Wildlife Refuge System, etc. On the one hand, people actively manage their land to attract game / fowl enrichment, on the other hand there are pockets where populations can always recover.
Later (2008-2016) secretary of National Ocean and Atmosphere Administration (NOAA).
A functional response is the intake rate of a consumer as a function of food density.
\[f(V) = {aV \over 1+ahV}\]
Where \(a\) is attack rate, \(h\) is handling time.
This assumes a single predator species focusing on a single prey species, getting slowed down by processing time.
An assistant picked sandpaper discs off a tackboard (Holling 1951).
Fundamental foundations for predation theory ensued.
Here, we assume that when prey are very scarce, there are other options for predators, or non-linear reasons why they’re harder to find.
Total Numerical Prey Response = Prey eaten / predator X predator response
Total Proportional Prey Response = Total prey response / Prey abundance
This curve will (typically) peaks at some intermediate number, where:
1) - N. Predators is (already) high,2) - N. of Prey consumed per Predator is high (approaching asymptote),but
3) - Total N of prey is NOT too high.All of that is total mortality of prey. Now - what we need is proporional growth of prey. We have a good, time-tested model for that:
Where the two lines intersect, Growth = Removal , we’re at equilibrium. The question is … are these equilibria stable?
Follow the arrows!
As usual, super elegant theory … but what is it good for?
Big Q: Are herbivore populations controlled bottom-up (food limited) or top-down (predator limited).
Food regulates
Data aggregated from many sites with high and low moose and wolf densities.
Looks Type II-ish, which makes sense since moose is (often) the main prey of wolves, and wolves are (often) the main predator of moose.
Note: these data are only from Isle Royale (Lake Superior) long term study.
low productive environment (3) system more likely to be predator limited
more productive environment (1) more likely to be food-limited
a 2-state model is unlikely ….
… but the numeric response curve is shallow. Meaning a local system (when perturbed) can switch from one to the other.
Consistent with observed variability not just in space but in time.